The local linear approximation to the function $f$ at $x=1$ is $y=2x+8$. What is the value of $f(1)+f'(1)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $9$ (Choice B) B $10$ (Choice C) C $11$ (Choice D) D $12$
Solution: The local linear approximation of $f$ at $x=1$ is achieved using the equation of the line tangent to $f$ at $x=1$. In other words, $y=2x+8$ is the equation of the line tangent to the graph of $f$ at $x=1$. How can we use this to find $f(1)$ and $f'(1)$ ? Since the line is tangent to the graph of $f$ at $x=1$, we know two key facts about it: The line passes through the point $({1},{f(1)})$ The line's slope is ${f'(1)}$ The slope of $y={2}x+8$ is ${2}$. The $y$ -value that corresponds to $x={1}$ is $2({1})+8={10}$. Now we can find our answer: ${f(1)}+{f'(1)}={10}+{2}=12$